The Sabathé cycle, also called dual combustion or limited or mixed pressure or Trinkler or Seiliger, is a reference thermodynamic cycle for internal combustion engines in which combustion takes place partly at constant pressure and partly at constant volume.

The actual operating conditions of diesel engines differ markedly from those represented in the ideal Otto and Diesel cycles. For Diesel engines, the combustion process approximates a constant pressure transformation only in the case of exceptionally large and slow engines.

In diesel engines and under normal conditions, the actual diagram shows that combustion is carried out according to a process that approximates the combustion of a transformation at constant volume and another at constant pressure.

It can be affirmed that, in practice, the Otto and Diesel cycles are very close in shape, to the point that they can be considered as a particular case of the mixed cycle, in which part of the combustion is checked at a constant volume, and part, at constant pressure. This theoretical cycle is known as Sabathé mixed cycle.

## Description of the Sabathé mixed cycle

In this cycle, after the adiabatic compression phase 1-2, a constant volume combustion phase 2-3 occurs. During this combustion phase the amount of heat Q1 'is introduced and then, as in the Diesel cycle, a 3-4 phase of constant pressure combustion, in which course the amount of heat Q1' 'is introduced.

Then follow two successive phases, namely: one, of adiabatic expansion 4-5, and another, of subtraction, at a constant volume 5-1, of the amount of heat Q2.

Therefore, the total amount of heat introduced is worth:

Q_{1}=Q_{1}'+Q_{1}''

Remembering the above, regarding the Otto and Diesel cycles, we can write:

Q _{1} '= C _{v} (T _{3} -T _{2} )

Q _{1} '' = C _{p} (T _{4} -T _{3} )

Q _{2} = C _{v} (T _{5} -T _{1} )

Thus, the ideal thermal efficiency of the theoretical Sabathé cycle is worth:

h _{e} = (heat supplied - heat subtracted) / heat supplied

For the 2-3 transformation of combustion at constant volume we have:

And for the 3-4 transformation of combustion at constant pressure:

For the adiabatic transformations 1-2 of compression and 4-5 of expansion we will use, respectively, the formulas

from which we get:

and being V _{3} = V _{2} ; V _{5} = V _{1}

It can be written:

Replacing these expressions in those of the ideal thermal efficiency, results:

Start with the relationship between pressure P3 at the end and pressure P2 at the beginning of the constant volume combustion phase - which we will call "constant volume combustion ratio" -, and remembering that:

The final expression of **ideal thermal performance of the Sabathé theoretical cycle is obtained** :

At equal compression ratio r, the performance of the mixed cycle is intermediate between that of the Otto cycle and that of the Diesel cycle. If the heat supplied is increased to a constant volume, that is, between points 2 and 3, and that supplied at a constant pressure between points 3 and 4 is reduced, the thermal efficiency is close to that of the Otto cycle. If, on the contrary, the heat supplied is reduced to a constant volume and the one corresponding to constant pressure is increased, the performance of the mixed cycle approximates that of the Diesel cycle.